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Theorem csbnest1g 2929
Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
csbnest1g  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )

Proof of Theorem csbnest1g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 2910 . . . 4  |-  F/_ x [_ y  /  x ]_ C
21ax-gen 1354 . . 3  |-  A. y F/_ x [_ y  /  x ]_ C
3 csbnestgf 2926 . . 3  |-  ( ( A  e.  V  /\  A. y F/_ x [_ y  /  x ]_ C
)  ->  [_ A  /  x ]_ [_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
42, 3mpan2 409 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
5 csbco 2889 . . 3  |-  [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ B  /  x ]_ C
65csbeq2i 2904 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ A  /  x ]_ [_ B  /  x ]_ C
7 csbco 2889 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C
84, 6, 73eqtr3g 2111 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259    e. wcel 1409   F/_wnfc 2181   [_csb 2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881
This theorem is referenced by:  csbidmg  2930
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